函数在一点的可导性与在该点附近的导数的关系

本文通过研究函数在一点的可导性与函数在该点附近的导数的关系,得出了函数在一点可导的一个充要条件,并通过举例说明了它在研究分段函数在其分段点的可导性方面的运用。     

Picture This: Multimodal representations of prospective teachers' metaphors about teachers and teaching

Metaphors are often used to help prospective teachers' articulate ideas and knowledge about teachers and teaching. This article reports the results of a study focused on a multimodal project completed by prospective teachers (N = 52) for the express purpose of identifying, capturing, displaying, and articulating their metaphorical definition(s) of teachers and teaching. Findings suggest that multimodality offered varied and nuanced ways for participants to represent and share metaphors. Findings also suggest that offering prospective teachers with opportunities to consider and articulate metaphorical ideas through multimodal means may further contribute to their development as they learn to teach.     

Interconnecting Content and Community: A Qualitative Study of Secondary Mathematics Teachers

The publication of the National Council ofTeachers of Mathematics initial Standards(1989) has acted as a catalyst to beginreforming the way mathematics is taught in theUSA. However, the literature regarding reformmovements suggests that changing oureducational systems requires overcoming manybarriers and is thus difficult to achieve.Reform in mathematics education, like reformmovements in other areas of education, has thusbeen slow to take hold. One structure that hasbeen shown to support educational reform,particularly instructional reform, has beenteacher community. This paper discusses aprofessional development intervention thatattempted to start a professional communityamong a group of secondary mathematics teachersthrough in-service work on mathematical problemsolving and technology. The results of thisstudy suggest that the use of mathematicalcontent explorations in professionaldevelopment settings provides a means to helpmathematics teachers build professionalcommunities. Together, these two components –mathematical content explorations and teachercommunity – provided these secondarymathematics teachers with a strong foundationfor engaging in the reform of their mathematicsclasses.     

初中农村教师数学知识状况及来源的调查研究——以勾股定理为例

初中农村教师所给出勾股定理的证明方法或思路都是正确的,但是这些证明方法或思路大多来自于教材;"教科书"和"自身的教学经验和自我反思"是初中农村教师发展数学知识的两个主要来源,其次是"作为学习者的经验",接下来是"专业书刊"、"和同事的日常交流"、"课堂听课和教研活动"、"教学辅导资料"、"在职培训"、"网络资源"和"职前教育".促进初中农村教师数学知识发展需要做到:充分利用发展数学知识的一些常见途径,不断扩充发展数学知识的途径.     

Teacher change: ideas emerging from a project for the teaching of university mathematics

This paper reports the changes that occurred in the didactic approaches of three professors who participated in a project intended to develop new ways of teaching mathematics to second year university students. An enactivist perspective is used to address the process of change that emerged as a result from interactions during project meetings. We describe changes in the participants' actions by looking at data obtained from the meetings and the classrooms. Teachers were able to ‘see more’ and modify their teaching practices incorporating a more open and flexible approach in accordance to their structural state which depended on their previous history. Therefore, the results varied. It was possible to observe, however, similar changes in all members of the group which included the use of vocabulary from learning theories and the inclusion of in-depth reflections on teaching and learning.     

引导学生欣赏与发现数学美——以极限教学为例

数学既求真又求美.数学求真在于用数量关系和空间图形表达世界;数学求美则在于用美丽的图形、精炼的语言、简练的定理、公式给人以视角和精神上的享受.在数学教学中,应挖掘教材人文因素以培养学生人文精神,充分利用数学史料以提升学生人文品质,在解题中揭示哲理以丰富学生人文素养,利用教师人格魅力以提升学生人文品位,建设数学课堂文化以充实学生人文内涵.     

Mathematics for teaching: A form of applied mathematics

In this article we elaborate a conceptualisation of mathematics for teaching as a form of applied mathematics (using Bass's idea of characterising mathematics education as a form of applied mathematics) and we examine implications of this conceptualisation for the mathematical preparation of teachers. Specifically, we focus on issues of design and implementation of a special kind of mathematics tasks whose use in mathematics teacher education can support the development of knowledge of mathematics for teaching. Also, we discuss broader implications of the article for mathematics teacher education, including implications for mathematics teacher educators' knowledge for promoting mathematics for teaching.     

Teachers and `Street Children': On Becoming a Teacher of Mathematics

The phenomenon of `street children' where youngpeople, for various reasons, live on thestreets of towns and cities is found all overthe world in varying degrees and forms. InSouth Africa, one approach to take care aboutthe plight of these children has been to set upand run what are referred to as `streetshelters'. One such street shelter, the onlyone exclusively for girls in the city ofDurban, is Tennyson House. In this paper Idescribe an innovative outreach programmeintegrated with a university curriculum inwhich a group of pre-service teachers takingmathematics education as a major were involvedin teaching mathematics to girls at TennysonHouse. From the vantage point of a mathematicsteacher educator in the programme, I describeand reflect on what was experienced and learnedfrom the intervention in terms of threeaspects: learning about learners; learningabout teaching (mathematics) and learning aboutrelationships.     

The impact of Specialist School status: a case study of two contrasting mathematics and computing colleges

The research examines the range of effects of obtaining Specialist School status in two contrasting mathematics and computing colleges, concentrating on the mathematics department. The positive impact of a wider range of technology was evident in both schools although the inherent pedagogical perspectives within each mathematics department remained fixed. Some definite tensions were evident in both schools – timetabling difficulties mitigating against maintaining and strengthening partnerships, a growing assumption that, having obtained Specialist School status, all mathematics staff were deemed to be ‘experts’. Key subject personnel in both schools obtained promotion within two years; each left a significant ‘gap’ in provision and development and raised issues of ‘succession management’ which had been unforeseen.     

幼儿园教师数学学科教学知识现状研究——以一位熟手型教师为例

采用课堂观察法和访谈法研究一位熟手型幼儿园教师的数学学科教学知识现状,结果发现：（1）数学内容知识方面,教师重视幼儿数学内容知识学习的衔接性和全面性。（2）数学教学策略知识方面,教师在教学中鼓励幼儿自由操作和探索,常采用直观材料演示的教学策略。（3）幼儿的知识方面,教师更为关注幼儿已有数学知识的经验准备,对本班幼儿有较全面的了解。（4）数学教学情境知识方面,常以创设游戏情境的方式开展教学活动,能为幼儿的数学知识学习创设问题情境。     

Pedagogies of experience: a case of the African American male teacher

Numerous scholars have illustrated how African American teachers’ past experiences provide them a philosophical vision committed to teaching for social and educational change for African American students. This article draws from this body of work by looking at the diverse ways five African American male teachers used their past experiences to shape their vision for working with African American male students. However, this article also extends this body of work by illustrating that while the teachers in this study had similar commitments to working with African American males, their varied life experiences and social locations provided them different sources of practical and philosophical knowledge. These findings illustrate the diversity of experiences these teachers drew from, thus troubling the common sense discourses that essentialize African American male teachers as one-dimensional monolithic role models.     

Developing assessment literacy through approximations of practice: Exploring secondary mathematics pre-service teachers developing criteria for a rich quadratics task

This study draws on a theoretical model for teacher assessment literacy to investigate the process of how secondary mathematics pre-service teachers (PSTs) might develop theirs through experiencing an approximation of practice. Sixty PSTs participated in designing and then refining assessment criteria for a rich open-ended quadratics task using real student example responses. Data analysis examined variations in the PSTs’ attention to quadratic features and mathematical language as assessment criteria for the task throughout the activity. This paper discusses evidence suggesting that the collaborative analysis of example student responses can make certain, but not all, assessment criteria salient to PSTs.     

Decisions to enter and continue in the teaching profession: Evidence from a sample of U.S. secondary STEM teacher candidates

Given its prevalence and cost it is imperative to identify predictors of early career teacher turnover intentions and behavior. During their final year as education majors, 311 US, STEM Secondary Education students rated their student teaching experience, the strength of their teacher identity, and their intent to enter the teaching profession. Within 1–3 years after graduating 191 of them reported whether they remained in the teaching profession. One's identity as a teacher, as well as the perceived quality of student teaching experiences, predicted both intent and actual entry into the teaching profession. Furthermore, teacher identity mediated the relationship between student teaching satisfaction and outcomes.     

The centrality of Aboriginal cultural workshops and experiential learning in a pre-service teacher education course: a regional Victorian University case study

This paper discusses a cross-cultural pedagogical approach, couched in a theory–practice nexus, used at a Victorian regional university to guide non-Indigenous pre-service teachers’ (PSTs) engagement with Aboriginal and Torres Strait Islander perspectives and cultures. We have drawn on qualitative and statistical data, and current issues in Australian and international literature, to explore the relevance and success suggested by data from this cross-cultural pedagogical approach, in particular the notion of teacher ethnicity in racialised spaces. In doing so, we have addressed recent sentiments about a lack of quantitative and qualitative research that explores inclusion of Aboriginal and Torres Strait Islander content and discussions of ways in which tertiary educators construct and influence teachings about Aboriginal and Torres Strait Islander cultures. It is anticipated that this paper will generate further dialogue and research-based evidence on ways in which other tertiary education providers may draw on cross-cultural theories to guide inclusion of Aboriginal and Torres Strait Islander content and perspectives in PST education courses.     

Dealing with gender in the classroom: a portrayed case study of four teachers

This paper emerges from a qualitative case study that takes place in a US preschool setting and explores teachers’ influences on the construction of children’s gender identities. According to postmodern theories of gender, identity is constructed and constituted through social interactions and performances. This study focuses on the gender identities constructed as preschool teachers and students interact and learn. In this portrayed case study, we combine the methodological elements of case study and portraiture. Portraiture methods enhance the traditional case study process, authorising a more considered presentation of participants and context. Four teachers at one preschool setting are portrayed. Some of the findings include the teachers’ awareness of the importance of their interactions with students and the impact they have on students’ gender performances. We also suggest that teachers need more resources and self-awareness regarding their own gender performances; teachers’ self-knowledge may be important in disrupting gender-stereotyped teaching and social constructions, and in creating more inclusive learning spaces for all students.     

Teacher creativity within the current education system: a case study of the perceptions of primary teachers

This article reports on a case study that examined the views of primary teachers in one school about their ability to be creative practitioners within the current education system. This research was conducted following the recent emphasis that teacher creativity has been given in government debate. Semi-structured interviews were carried out with 10 members of teaching staff. The main findings presented suggest that the pressures of the curriculum, and the targets and objectives laid down within it, severely restrict and constrain the creative practices that the teachers in this school feel able and willing to engage in. Perceptions of the relevance and value of teacher creativity are also discussed.     

Understanding the co‐construction of inquiry practices: A case study of a responsive teaching environment

We set out to understand how different instantiations of inquiry emerged in two different years of one elementary teacher's classroom. Longitudinal observations from Mrs. Charles' 5th grade science classroom forced us to carefully and deliberately consider who exactly was responsible for the change in the class activities and norms. We provide empirical evidence to show how a focus on the teacher can easily overlook the complex dynamics of the classroom. The data reveal that students had a substantive and generative role in the class's arrival at the different instantiations of scientific inquiry—the nature and form of inquiry—that were constructed each year. We argue that, in an environment where a teacher carefully attends and responds to student thinking, the nascent resources students have for reasoning about phenomena can affect not only the conceptual ideas that emerge, but also influence what inquiry activities or practices become established as normative and productive over time. Our work with Mrs. Charles illuminates an important methodological concern with research on teacher development as well as the construct of teacher learning progressions; research accounts that focus primarily on the teacher may overlook the classroom norms that are negotiated between teacher and student, and thereby provide an incomplete portrayal of the teacher's activity within one classroom and the teacher's progress across multiple years. © 2012 Wiley Periodicals, Inc. J Res Sci Teach 49: 429–464, 2012     

‘Them and us’: Teachers' and tutors' experiences of a ‘new’ professional development course in Greece

The research reported in this paper compares the experiences of twenty‐two secondary teachers and twelve university tutors as they embark on a course of a ‘new’ in‐service professional development programme in Greece, in relation to adult learning. It harnesses a qualitative methodology and draws upon a set of ideas that cohere under the rubric programme development to build up a heuristic guiding research technique and analysis procedure. The analysis focuses on two key parameters of programme development for adults i.e. climate and planning, and on their associated elements. The findings identify the ‘them’ and ‘us’ image that continues to perpetuate the ‘new’ courses and indicate some tentative points that might have implications for the design of the in‐service training of teachers.     

Prerequisites for the successful group mentoring of first-year university students: a case study

In this article we present a case study on a group mentoring practice proven successful in earlier studies in terms of student self-regulation and collaboration. The purpose of our study was to uncover the factors behind the success by interviewing the mentor teachers. The findings showed that the group mentoring focused on four main themes: (a) promoting social relationships, (b) providing personal support, (c) providing study guidance, and (d) strengthening the agency of students. The findings suggest that the leading ethos of group mentoring was related to a humanistic approach to mentoring. However, successful mentoring also seemed to require both situated apprentice and critical constructivist perspectives in addition to the humanistic approach. It is concluded that the combination of multiple goals and contents in group mentoring is the main contributing factor behind the success of the mentoring model examined. Additionally, the teachers reported a variety of positive impacts of group mentoring on teachers’ work.     

Becoming a lecturer in further education in England: the construction of professional identity and the role of communities of practice

Further education colleges in England offer a wide range of post‐school education and training provision. Recently they have undergone major transformations that have resulted in considerable changes to the work of those teaching in them. In this paper we examine how cultures of learning and teaching in colleges are affected and how the nature of professional identity has changed. The paper considers the formation of professional identity amongst a group of trainee lecturers completing a one‐year full‐time teacher‐training course at a university in the English Midlands. Lave and Wenger's work on apprenticeship to communities of practice is used to examine the effect of trainees' teaching placement on the development of professional identity. Rather than identifying effective processes of increasing participation in existing communities of practice, a strong sense of marginalisation and alienation amongst trainees was observed. The paper argues that this is detrimental both to trainees and experienced lecturers if they are to actively engage in building new forms of professionalism for the future.